Geometry

Question Analysis

The main focus is leveraging the properties of 45 - 45 - 90 and 30 - 60 - 90 triangles to determine the value of . It requires correctly applying the side - length relationships of these special triangles.

 

Key Concept Explanation

45 - 45 - 90 triangle: In a 45 - 45 - 90 triangle, if the leg length is , the hypotenuse length is .

30 - 60 - 90 triangle: In a 30 - 60 - 90 triangle, if the short leg (opposite 30°) has length , the long leg (opposite 60°) has length , and the hypotenuse (opposite 90°) has length .

 

Step - by - Step Solution

1. Analyze the 30 - 60 - 90 triangle:

Since the side with length is opposite the 60° angle, and the common side is the short leg. Let the short leg be . Using the formula for the long leg , we solve for :

2. Analyze the 45 - 45 - 90 triangle:

The short leg

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